More by Anders J. Frankild

Abstract

The main result asserts that a local commutative Noetherian ring is
Gorenstein, if it possesses a non-zero cyclic module of finite
Gorenstein injective dimension. From this follows a classical result
by Peskine and Szpiro stating that the ring is Gorenstein, if it
admits a non-zero cyclic module of finite (classical) injective
dimension. The main result applies to local homomorphisms of local
rings and yields the next: if the source is a homomorphic image of a
Gorenstein local ring and the target has finite Gorenstein injective
dimension over the source, then the source is a Gorenstein ring.
This, in turn, applies to the Frobenius endomorphism when the local
ring is of prime equicharacteristic and is a homomorphic image of a
Gorenstein local ring.